1

Projection

2

Model Transform

Viewing Transform

ModelviewMatrix

worldcoordinates

Pipeline Review

Focus of this lecture

3

Review (Lines in R2)

2121

2121

,

0 0

:lines twoofon intersecti asPoint

0 0

:line a determine points Two

0

equation line sHomogeneou

),,(),,,(

00

llplpandlp

ppllpandlp

lp

wvupcbal

cwbvaucbyax wv

wu yx

4Parallel Projection

Projection (R2)

viewpoint

viewline

bacabcbac

bacabccba

bacabcbac

bacabccba

5

Perspective Projection

~

~

6

Parallel Projection

~

~

7

Projection (R3)

See handout for proof!

8

ExampleVertices (0,0,0), (2,0,0), (2,3,0), (0,3,0) (1,1,1), (1,2,1)

Parallel projection: onto z = 0 planev = (0,0,1,0)T, n = (0,0,1,0)T

9

Vertices (0,0,0), (2,0,0), (2,3,0), (0,3,0) (1,1,1), (1,2,1)

Perspective projection: onto z = 0 plane from viewpoint (1,5,3)v = (1,5,3,1)T, n = (0,0,1,0)T

10

321

321

321

,,ˆ

,,ˆ

,,

ssss

rrrr

qqqO

p’ p” O

Viewplane Coordinate Mapping

11

Determine Viewplane Transform by Homogeneous Transformation

144313 pVp133414 pKp

001

100

010

001333

222

111

srq

srq

srq

K4×3

12

144313 pVp133414 pKp

pIpKp LL

L: left inverse of K

13

ExampleViewplane origin (1,2,0) u-axis (3,4,0) v-axis (-4,3,0)

14

Orthographic Projection

• Def: direction of projection viewplane

0,,, :viewpoint

,,, : vectorviewplane

321

4321

nnnv

nnnnn

v

n

… is a parallel projection

)(000

)(

)(

)(

23

22

21

4322

213231

423223

2121

41312123

22

nnn

nnnnnnnn

nnnnnnnn

nnnnnnnn

15

Definitions• Direction cosine (ref)

• Foreshortening ratio= (length of projected segment)/(length of original segment)

1

cos,cos,cos

222

nml

A

An

A

Am

A

Al zyx

16

Theorem

• If the direction cosines of the plane normal (in world coordinate system) are n1, n2, and n3, the foreshortening ratios in the x-, y-, and z- directions are (n2

2 + n32)1/2, (n1

2 + n32)1/2, and (n1

2 + n22)1/2,

respectively.• Front, side, top views: n =

(1,0,0,0), (0,1,0,0), or (0,0,1,0) as in engineering drawings

17

Types of Orthographic Projections

• Axonometric projections: attempts to portray general 3D shape– Isometric projection: all foreshortening ratio are

the same – Dimetric projection: exactly two are the same– Trimetric projection: all foreshortening ratio are

different

18

Axonometric Projections

3

636

36

31

31

31

,,

0,,,

f

n 3

2232

322

31

37

31

,,

0,,,

f

n 75

747526

32

153

1537

33

,,

0,,,

f

n

Isometric Dimetric Trimetric

f: foreshortening ratios

19

Example (Dimetric)

1000

0

0

0

0

98

97

91

97

92

97

91

97

98

31

37

31

M

n

TT

TT

TT

TT

zz

yy

xx

oo

1' 1100

1' 1010

1' 1001

1000' 1000

98

97

91

97

92

97

91

97

98

322

32

322

zo

yo

xo

20

21

Oblique Projection

• A particular parallel projection where direction of projection is not perpendicular to viewplane

v

n

Oblique projection not available in

OpenGL

22

Cavalier Projection

Lines viewplane have f = 1Planar faces viewplane appear thicker

v

/4n

Properties:

viewplan

e

23

Cabinet Projection

To overcome ‘thickness’ problem, choose f viewplane to be 1/2

Properties:

= arccot(2)

v

n

24

Perspective Projection

• A perspective projection maps parallel lines in the space to parallel lines in the viewplane IFF the lines are parallel to the viewplane.

25

Otherwise, they meet

26

Vanishing Point

• Suppose (xi, yi, zi) i =1,2,3 are a set of mutually perpendicular vectors. The viewplane normal (n1, n2, n3) of a perspective projection can be perpendicular to (a) none (b) one (c) two of the vectors.

(a) (b) (c)

n

n

n

27

Vanishing Point

• If a perspective projection maps a point-at-infinity (x,y,z,0) to a finite point (x’,y’,z’,1) on the viewplane, the lines in the direction (x,y,z) appear as lines converging to point on the (Cartesian) viewplane. The point (x’,y’,z’) is called the vanishing point in the direction (x,y,z).

28

Three-point perspective

Two-point perspective

One-point perspective

Vanishing point

29

IMAGE FORMATION – Perspective Imaging

Image courtesy of C. Taylor

“The Scholar of Athens,” Raphael, 1518

30

Example

• Determine (and verify it is indeed so) the vanishing point of an OpenGL setting.

Eye = [15,0,0] Eye = [15,0,15]

31

Numeric Example

TT

TT

TT

T

T

T

M

M

M

Verify

IvnvnM

n

v

003100010

1160150100

1150160001

:

30101

1516015

00310

1515016

1101

115015

4

How about (1,0,1,0)?

Viewpoint (15,0,15,1)Viewplane: x + z + 1 = 0

32

Summary

• Projection– Parallel projection– Perspective projection

• Parallel projection– Orthographic

• Isometric• Dimetric• Trimetric

– Oblique• Cavalier• Cabinet

• Perspective projection– Three-point

perspective

– Two-point perspective

– One-point perspective

Understand how they are

differentiated

33

Fig. 8. Constructing a perspective image of a house. (a) Drawing the floor plan and defining the viewing conditions (observer position and image plane). (b) Constructing a perspective view of the floor. (c) A reference height (in this case the height of an external wall) is drawn from the ground line and the first wall is constructed in perspective by joining the reference end points to the horizontal vanishing point v2. (d) All four external walls are constructed. (e) The elevations of all other objects (the door, windows and roofs) are first defined on the reference segment and then constructed in the rendered perspective view.

34

Exercise

• Hand sketch a perspective drawing of a house

• Use Maxima to compute 2-point perspective projection, setting viewplane coordinate system

35

Cross Ratio

The cross-ratio of every set of four collinear points shown in this figure has the same value

Cross ratio is preserved in projective geometry(ratio is NOT preserved)

z1z2 z3 z4